# Properties

 Degree 56 Conductor $2^{56} \cdot 3^{28}$ Sign $1$ Motivic weight 15 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 1.34e4·4-s − 1.90e6·9-s + 1.24e8·13-s − 2.73e8·16-s + 3.53e11·25-s − 2.57e10·36-s + 3.86e11·37-s + 5.72e13·49-s + 1.67e12·52-s − 1.63e13·61-s − 9.05e11·64-s + 1.58e14·73-s + 3.08e13·81-s − 8.24e14·97-s + 4.77e15·100-s + 2.24e13·109-s − 2.37e14·117-s − 5.78e16·121-s + 127-s + 131-s + 137-s + 139-s + 5.22e14·144-s + 5.20e15·148-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 0.411·4-s − 0.132·9-s + 0.550·13-s − 0.255·16-s + 11.5·25-s − 0.0546·36-s + 0.668·37-s + 12.0·49-s + 0.226·52-s − 0.667·61-s − 0.0257·64-s + 1.67·73-s + 0.149·81-s − 1.03·97-s + 4.77·100-s + 0.0117·109-s − 0.0731·117-s − 13.8·121-s + 0.0338·144-s + 0.275·148-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(16-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{28}\right)^{s/2} \, \Gamma_{\C}(s+15/2)^{28} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$56$$ $$N$$ = $$2^{56} \cdot 3^{28}$$ $$\varepsilon$$ = $1$ motivic weight = $$15$$ character : induced by $\chi_{12} (1, \cdot )$ primitive : no self-dual : yes analytic rank = 0 Selberg data = $(56,\ 2^{56} \cdot 3^{28} ,\ ( \ : [15/2]^{28} ),\ 1 )$ $L(8)$ $\approx$ $47.9769$ $L(\frac12)$ $\approx$ $47.9769$ $L(\frac{17}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 56. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 55.
$p$$F_p$
bad2 $$1 - 1.34e4T^{2} + 4.55e8T^{4} - 8.93e12T^{6} + 2.03e18T^{8} + 4.79e21T^{10} + 1.11e27T^{12} - 2.66e30T^{14} + 1.19e36T^{16} + 5.52e39T^{18} + 2.51e45T^{20} - 1.18e49T^{22} + 6.50e53T^{24}+O(T^{26})$$
3 $$1 + 1.90e6T^{2} - 2.72e13T^{4} + 3.17e21T^{6} - 3.98e28T^{8} + 1.04e35T^{10} + 8.33e42T^{12} - 1.24e50T^{14} + 1.71e57T^{16}+O(T^{17})$$
good5 $$1 - 3.53e11T^{2} + 6.26e22T^{4} - 7.34e33T^{6} + 6.36e44T^{8} - 4.34e55T^{10}+O(T^{12})$$
7 $$1 - 5.72e13T^{2} + 1.60e27T^{4} - 2.97e40T^{6} + 4.12e53T^{8}+O(T^{10})$$
11 $$1 + 5.78e16T^{2} + 1.68e33T^{4} + 3.26e49T^{6} + 4.73e65T^{8}+O(T^{9})$$
13 $$1 - 1.24e8T + 8.15e17T^{2} - 9.32e25T^{3} + 3.33e35T^{4} - 3.56e43T^{5} + 9.16e52T^{6} - 9.27e60T^{7}+O(T^{8})$$
17 $$1 - 4.15e19T^{2} + 8.82e38T^{4} - 1.26e58T^{6}+O(T^{8})$$
19 $$1 - 2.28e20T^{2} + 2.67e40T^{4} - 2.13e60T^{6}+O(T^{7})$$
23 $$1 + 3.76e21T^{2} + 7.24e42T^{4} + 9.47e63T^{6}+O(T^{7})$$
29 $$1 - 1.60e23T^{2} + 1.27e46T^{4} - 6.62e68T^{6}+O(T^{7})$$
31 $$1 - 4.80e23T^{2} + 1.14e47T^{4} - 1.78e70T^{6}+O(T^{7})$$
37 $$1 - 3.86e11T + 3.80e24T^{2} - 6.10e35T^{3} + 7.31e48T^{4} + 2.36e59T^{5}+O(T^{6})$$
41 $$1 - 1.35e25T^{2} + 1.09e50T^{4}+O(T^{6})$$
43 $$1 - 3.21e25T^{2} + 5.32e50T^{4}+O(T^{6})$$
47 $$1 + 1.55e26T^{2} + 1.20e52T^{4}+O(T^{6})$$
53 $$1 - 9.76e26T^{2} + 4.65e53T^{4}+O(T^{6})$$
59 $$1 + 7.41e27T^{2} + 2.72e55T^{4}+O(T^{6})$$
61 $$1 + 1.63e13T + 1.11e28T^{2} + 1.68e41T^{3} + 6.17e55T^{4} + 8.62e68T^{5}+O(T^{6})$$
67 $$1 - 2.13e28T^{2} + 2.65e56T^{4}+O(T^{6})$$
71 $$1 + 8.14e28T^{2} + 3.23e57T^{4}+O(T^{6})$$
73 $$1 - 1.58e14T + 1.22e29T^{2} - 1.74e43T^{3} + 7.80e57T^{4} - 1.03e72T^{5}+O(T^{6})$$
79 $$1 - 4.09e29T^{2} + 8.13e58T^{4}+O(T^{6})$$
83 $$1 + 9.72e29T^{2} + 4.72e59T^{4}+O(T^{5})$$
89 $$1 - 3.65e30T^{2} + 6.53e60T^{4}+O(T^{5})$$
97 $$1 + 8.24e14T + 1.02e31T^{2} + 8.41e45T^{3} + 5.35e61T^{4}+O(T^{5})$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{56} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}